Maximum Principle

PDEs · Axiom Academy

Understanding how harmonic functions achieve their extrema on boundaries 1. The Maximum Principle Statement Let u be a harmonic function (satisfying Δ u = 0) on a bounded domain Ω with boundary ∂Ω. Then: The animation below shows a harmonic function on a disk. Notice how the function surface never exceeds the boundary values—the "peak" is always on the edge. 2. Physical Intuition: Heat Seeks Equilibrium Think of a harmonic function as a steady-state temperature distribution . If the temperature were higher at some interior point than everywhere on the boundary, heat would flow outward from that point. But this contradicts the steady-state assumption—the temperature should be constant in time! The animation illustrates this: if we suppose there's an interior maximum, heat flux vectors point outward, creating a contradiction. 3. Proof Sketch: Mean Value Property The maximum principle follows from the mean value property of harmonic functions: The animation shows how averaging prevents interior maxima: 4. Strong vs. Weak Maximum Principle There are two versions of the maximum principle, each with different applications: The animation contrasts these two scenarios: 5. Application: Uniqueness of Solutions The maximum principle immediately gives us uniqueness for the Dirichlet problem: The animation shows how the difference function must be zero: The maximum principle extends to comparison principles for more general equations: The animation shows two functions being compared:

This is the written version of the interactive lesson above. See the full PDEs course.